Understanding Markov Chains: Examples and Applications

Description

This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples (gambling processes and random walks) are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters.  An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions. Front Matter....Pages I-IX Introduction....Pages 1-6 Probability Background....Pages 7-36 Gambling Problems....Pages 37-60 Random Walks....Pages 61-75 Discrete-Time Markov Chains....Pages 77-94 First Step Analysis....Pages 95-116 Classification of States....Pages 117-128 Long-Run Behavior of Markov Chains....Pages 129-148 Branching Processes....Pages 149-166 Continuous-Time Markov Chains....Pages 167-209 Discrete-Time Martingales....Pages 211-223 Spatial Poisson Processes....Pages 225-239 Reliability Theory....Pages 241-245 Back Matter....Pages 247-354